Search Results (3)

Cut sets, decomposition theorem and representation theorem have a great influence on the realization of the transformation of fuzzy sets and classical sets, and the single-valued neutrosophic multisets (SVNMSs) as the generalization of fuzzy sets, which cut sets, decomposition theorem and representation theorem have the similar effects, so they need to be studied in depth. In this paper, the decomposition theorem, representation theorem and the application of a new similarity measures of SVNMSs are studied by using theoretical analysis and calculations. The following are the main results: (1) The notions, operation and operational properties of the cut sets and strong cut sets of SVNMSs are introduced and discussed; (2) The decomposition theorem and representation theorem of SVNMSs are established and rigorously proved. The decomposition theorem and the representation theorem of SVNMSs are the theoretical basis for the development of SVNMSs. The decomposition theorem provides a new idea for solving the problem of SVNMSs, and points out the direction for the principle of expansion of SVNMSs. (3) Based on the decomposition theorem and representation theorem of SVNMSs, a new notion of similarity measure of SVNMSs is proposed by applying triple integral. And this new similarity is applied to the practical problem of multicriteria decision-making, which explains the efficacy and practicability of this decision-making method. The new similarity is not only a way to solve the problem of multi-attribute decision-making, but also contains an important mathematical idea, that is, the idea of transformation. Full article

In real-world diagnostic procedures, due to the limitation of human cognitive competence, a medical expert may not conveniently use some crisp numbers to express the diagnostic information, and plenty of research has indicated that generalized fuzzy numbers play a significant role in describing complex diagnostic information. To deal with medical diagnosis problems based on generalized fuzzy sets (FSs), the notion of single-valued neutrosophic multisets (SVNMs) is firstly used to express the diagnostic information in this article. Then the model of probabilistic rough sets (PRSs) over two universes is applied to analyze SVNMs, and the concepts of single-valued neutrosophic rough multisets (SVNRMs) over two universes and probabilistic rough single-valued neutrosophic multisets (PRSVNMs) over two universes are introduced. Based on SVNRMs over two universes and PRSVNMs over two universes, single-valued neutrosophic probabilistic rough multisets (SVNPRMs) over two universes are further established. Next, a three-way decisions model by virtue of SVNPRMs over two universes in the context of medical diagnosis is constructed. Finally, a practical case study along with a comparative study are carried out to reveal the accuracy and reliability of the constructed three-way decisions model. Full article

Based on the multiplicity evaluation in some real situations, this paper firstly introduces a single-valued neutrosophic multiset (SVNM) as a subclass of neutrosophic multiset (NM) to express the multiplicity information and the operational relations of SVNMs. Then, a cosine measure between SVNMs and weighted cosine measure between SVNMs are presented to measure the cosine degree between SVNMs, and their properties are investigated. Based on the weighted cosine measure of SVNMs, a multiple attribute decision-making method under a SVNM environment is proposed, in which the evaluated values of alternatives are taken in the form of SVNMs. The ranking order of all alternatives and the best one can be determined by the weighted cosine measure between every alternative and the ideal alternative. Finally, an actual application on the selecting problem illustrates the effectiveness and application of the proposed method. Full article